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Lorenzo Sadun

Publications and source records attributed to Lorenzo Sadun.

71 records · Page 4Linked to original sources
arXiv2005

Topological mixing for substitutions on two letters

We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $θ_2$ of the substitution matrix satisfies $|θ_2|\ne 1$. If $|θ_2|<1$, then (as is well-known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|θ_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|θ_2|=1$ is more delicate, and we only obtain some partial results.

math.DS↗
arXiv2005

Most stable structure for hard spheres

The hard sphere model is known to show a liquid-solid phase transition, with the solid expected to be either face centered cubic or hexagonal close packed. The difference in free energy between the two structures is very small and various attempts have been made to determine which one is the more stable. We contrast the different approaches and extend one.

cond-mat.stat-mech↗
arXiv2005

Periodic diffraction patterns for 1D quasicrystals

A simple model of 1D structure based on a Fibonacci sequence with variable atomic spacings is proposed. The model allows for observation of the continuous transition between periodic and non-periodic diffraction patterns. The diffraction patterns are calculated analytically both using ``cut and project'' and ``average unit cell'' methods, taking advantage of the physical space properties of the structure.

cond-mat↗
arXiv2004

The structure of the hard sphere solid

We show that near densest-packing the perturbations of the HCP structure yield higher entropy than perturbations of any other densest packing. The difference between the various structures shows up in the correlations between motions of nearest neighbors. In the HCP structure random motion of each sphere impinges slightly less on the motion of its nearest neighbors than in the other structures.

cond-mat.stat-mech↗
arXiv2003

Oort's conjecture for A_g

We prove the conjecture of Oort that a compact subvariety of the moduli space of principally polarized Abelian varieties of genus g has codimension strictly greater than g, for g > 2, in characteristic zero

math.AG↗
arXiv2002

Tiling Spaces are Inverse Limits

Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology. Then Omega is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Gamma. This result extends previous results of Anderson and Putnam, of Ormes, Radin and Sadun, of Bellissard, Benedetti and Gambaudo, and of Gähler. In particular, the construction in this paper is a natural generalization of Gähler's.

math.DS↗
arXiv2002

Spherically symmetric solutions of a boundary value problem for monopoles

In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.

math.DG↗
arXiv2000

Fredholm Indices and the Phase Diagram of Quantum Hall Systems

The quantized Hall conductance in a plateau is related to the index of a Fredholm operator. In this paper we describe the generic ``phase diagram'' of Fredholm indices associated with bounded and Toeplitz operators. We discuss the possible relevance of our results to the phase diagram of disordered integer quantum Hall systems.

math-ph↗
arXiv2000

A Homeomorphism Invariant for Substitution Tiling Spaces

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Cech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.

math.DS↗
arXiv1998

On Angles Whose Squared Trigonometric Functions are Rational

We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector space over Q generated by these angles. Geodetic angles and rational linear combinations of geodetic angles appear naturally in Euclidean geometry; for illustration we apply our results to equidecomposability of polyhedra.

math-ph↗
arXiv1997

Some Generalizations of the Pinwheel Tiling

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but CAN be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite.

math.GR↗
arXiv1997

An Algebraic Invariant for Substitution Tiling Systems

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.

math.DS↗
arXiv1997

Parallel Connections Over Symmetric Spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.

dg-ga↗
arXiv1997

Simple Type is Not a Boundary Phenomenon

This is an expository article, explaining recent work by D. Groisser and myself [GS] on the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. The presentation is intended to complement [GS], presenting the essential ideas rather than the analytical details. It is shown that the boundary region of moduli space contributes 6/64 of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to a statement about the interior of moduli space, namely that the interior of the (k+1)st ASD moduli space, intersected with two representatives of (4 times) the point class, be homologous to 58 copies of the (k)-th moduli space. This is peculiar, since the only known embeddings of the (k)-th moduli space into the (k+1)st involve Taubes patching, and the image of such an embedding lies entirely in the boundary region.

dg-ga↗
arXiv1997

Simple Type and the Boundary of Moduli Space

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. Using a geometric representative of μ(pt), the boundary region of moduli space contributes 6/64 of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the k+1st ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the k-th moduli space. This is peculiar, since the only known embeddings of the k-th moduli space into the k+1st involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region. When using de Rham representatives of mu(pt), the boundary region contributes 1/8 of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is surprising but not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.

dg-ga↗
arXiv1996

Subgroups of SO(3) Associated with Tilings

We give a thorough analysis of those subgroups of SO(3) generated by rotations about perpendicular axes by 2π/p and 2π/q. A corollary is that such a group is the free product of the cyclic groups of rotations about the separate axes if and only if p,q\ge 3 and are both odd. These groups are naturally associated with a family of hierarchical tilings of Euclidean 3-space.

math.GR↗